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Search: id:A118921
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| A118921 |
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Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n having first return to the x-axis at (2k,0) (n,k>=1). (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)). |
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+0
2
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| 2, 4, 2, 12, 4, 4, 40, 12, 8, 10, 140, 40, 24, 20, 28, 504, 140, 80, 60, 56, 84, 1848, 504, 280, 200, 168, 168, 264, 6864, 1848, 1008, 700, 560, 504, 528, 858, 25740, 6864, 3696, 2520, 1960, 1680, 1584, 1716, 2860, 97240, 25740, 13728, 9240, 7056, 5880, 5280
(list; table; graph; listen)
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OFFSET
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1,1
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COMMENT
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Row sums are the central binomial coefficients (A000984). T(n,0)=2*A028329(n-1). Sum(k*T(n,k),k>=1)=2^(2n-1) (A004171). For returns to the x-axis arriving from above, see A039599.
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FORMULA
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T(n,k)=2*binomial(2k-2,k-1)*binomial(2n-2k,n-k)/k. G.f.=G(t,z)=[1-sqrt(1-4tz)]/sqrt(1-4z).
T(n+1,k+1)=2*(n-k+1)*A078391(n,k), n>=0, k>=0 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 13 2006
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EXAMPLE
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T(3,2)=4 because we have uudd|ud, uudd|du, dduu|ud, and dduu|du (first return to the x-axis shown by | ).
Triangle starts:
2;
4,2;
12,4,4;
40,12,8,10;
140,40,24,20,28;
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MAPLE
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T:=(n, k)->2*binomial(2*k-2, k-1)*binomial(2*n-2*k, n-k)/k: for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000984, A028329, A004171, A039599.
Sequence in context: A097692 A118920 A125755 this_sequence A121799 A078034 A138770
Adjacent sequences: A118918 A118919 A118920 this_sequence A118922 A118923 A118924
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 06 2006
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